Through comprehensive density functional calculations, the crystallographic, magnetic and electronic properties of $Na_xCoO_2$ ($x$ = 1, 0.875, 0.75, 0.625 and 0.50) were investigated. We found that all Na ions in $NaCoO_2$ and $Na_{0.875}CoO_2$ shar
e the basal coordinates with O ions. However, as $x$ decreases, some of Na ions move within the basal plane in order to reduce the in-plane Na$-$Na electrostatic repulsion. Magnetically, there was strong tendency for type A antiferromagnetism in the $Na_{0.75}CoO_2$ system, while all other Na deficient systems had a weaker ferromagnetic tendency. The results on magnetism were in excellent agreement with the experiments.
In this work, we investigated the behaviour of Sb dopants in $Na_{x}CoO_{2}$ for Na concentrations of $x = 0.75, 0.875$ and $1.00$ by density functional theory. We chose $Na_{x}CoO_{2}$ with higher Na concentration of $x > 0.75$ because it has excess
ively higher thermo-power thus it is appealing for practical applications. The rationale for choosing Sb was its exceedingly higher atomic mass than all elements of the host crystal which enable Sb to rattle phonons considerably.
